Year 8AC9M8M05

Rates, Speed and Unit Conversion

Solve problems involving rates, including speed and density, and convert between units of rate.

A rate compares two different units. Get the units right and the sums look after themselves.

Builds onYr 7 · Ratios and proportion

Why this works

A rate is a division, and the word 'per' tells you exactly what to divide by what.

'Per' means 'for each one'. 60 km per hour means 60 km travelled for each hour.

\text{speed} = \frac{\text{distance}}{\text{time}}

Because it is a division, the same triangle relationship applies as with any formula of this shape.

d = s \times t \qquad t = \frac{d}{s}

Converting rates means converting both units. To go from km/h to m/s, change kilometres to metres and hours to seconds.

1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}}

That simplifies to a single conversion factor worth memorising.

\text{km/h} \div 3.6 = \text{m/s}

If you are ever unsure, write the units into the working. If the units of your answer come out as km/h when the question asked for m/s, you know immediately that something is wrong.

Worked example

e.g.

\text{A car travels } 210 \text{ km in } 2.5 \text{ hours. Find its average speed.}

Identify what you are given and what you want. $d = 210 \text{ km}, \quad t = 2.5 \text{ h}, \quad s = ?$
Choose the right arrangement of the formula. $s = \frac{d}{t}$
Substitute, keeping the units attached. $s = \frac{210 \text{ km}}{2.5 \text{ h}}$
Divide. $s = 84 \text{ km/h}$
Check the units are what the question asked for. ✓ $84 \text{ km/h}$

Common mistakes

Wrong

\text{Dividing time by distance}

Right

\text{speed} = \frac{\text{distance}}{\text{time}}

'Kilometres per hour' literally means kilometres divided by hours. Let the unit name tell you the order.

Wrong

2 \text{ h } 30 \text{ min} = 2.30 \text{ hours}

Right

2 \text{ h } 30 \text{ min} = 2.5 \text{ hours}

Minutes are sixtieths, not hundredths. 30 minutes is half an hour, so

2.5

. Only 60, not 100, makes a whole hour.

Wrong

\text{Converting km/h to m/s by} \times 1000

Right

\text{km/h} \div 3.6 = \text{m/s}

You must convert both units. Multiplying by 1000 handles kilometres to metres but ignores hours to seconds.

Practice

FluencyGet quick and accurate at the method.

150 \text{ km in } 3 \text{ h. Speed?}

Answer

50 \text{ km/h}

\text{At } 80 \text{ km/h for } 2.5 \text{ h. Distance?}

Answer

200 \text{ km}

\text{Travel } 300 \text{ km at } 60 \text{ km/h. Time?}

Answer

5 \text{ h}

\text{Convert } 72 \text{ km/h to m/s}

Answer

20 \text{ m/s}

\text{Convert } 15 \text{ m/s to km/h}

Answer

54 \text{ km/h}

\$18 \text{ for } 3 \text{ kg. Price per kg?}

Answer

\$6\text{/kg}

450 \text{ km on } 30 \text{ L. Fuel economy?}

Answer

15 \text{ km/L}

\text{Walk } 6 \text{ km in } 1 \text{ h } 30 \text{ min. Speed?}

Answer

4 \text{ km/h}

AppliedThe same maths, inside a real question.

A train leaves at 09:15 and arrives at 11:45, covering 200 km. What was its average speed?

Answer80 km/h. The journey took 2 h 30 min

= 2.5

h, and

200 \div 2.5 = 80

A runner completes 10 km at 12 km/h. How long did it take, in minutes?

Answer50 minutes.

t = 10 \div 12 = 0.8\overline{3}

h, and

0.8\overline{3} \times 60 = 50

min.

A car uses 8 L per 100 km. How much fuel is needed for a 350 km trip?

Answer28 L.

\tfrac{350}{100} \times 8 = 28

Which is faster: 90 km/h, or 26 m/s?

Answer26 m/s. Converting,

90 \div 3.6 = 25

m/s, which is less than 26 m/s.